Since
\(\Delta \psi _x+\omega ^{\prime }(\psi )\psi _x=0\;\;\mbox{in $D$},\) 
the assertion (i) for the domain \(D\)  and the inequalities
\(&&\psi _{xx}(0,y)>0\;\;\mbox{ for $0<y< y_0$},\;\; \psi _{xx}(\Lambda /2,y)<0\;\;\mbox{ for $0<y< y_1$} \\ &&\mbox{and}\;\;\psi _{xy}(x,0)>0\;\;\mbox{for}\;\;0<x<\Lambda /2\) 
follows from the maximum principle and Hopf lemma (see [1]} and [2]}). Let us prove that
\(\psi _{xx}(0,y_0)>0\;\;\;\mbox{and}\;\; \psi _{xx}(\Lambda /2,y_1)<0.\) 
Consider the first inequality. The tangent to the boundary curve at the point \((0,y_0)\)  is parallel to the \(x\) -axis and so the curve near this point can be described by the equation \(y=f(x)\)  with smooth \(f\)  and with \(f^{\prime }(0)=0\) . The Bernoulli equation on this piece of the curve has the form
\(\frac{1}{2}|\nabla \psi |^2+f(x)-h=Q\;\;\mbox{for $y=f(x)$}.\) 
Differentiating this relation with respect to \(x\)  and using that \(\psi (x,f(x))=0\) , which implies \(f^{\prime }=-\psi _x/\psi /y\) , we get
\(\partial _\nu (\psi _x)-\rho \psi _x=0\;\;\mbox{for $y=f(x)$}.\) 
Here \(\nu \)  is the outward unit normal and
\(\rho =\rho (x)=\frac{(1+\psi _x\psi _{xy}+\psi _y\psi _{yy})}{\psi _y(\psi _x^2+\psi _y^2)^{1/2}}\Big |_{y=f(x)}.\) 
By (REF ), (REF ) and (REF ) we conclude that the asymptotics of \(\psi _x\)  near the point \((0,y_0)\)  is \(\psi (x,y)=cx+O(r^2)\) , where \(c\ge 0\)  and \(r^2=x^2+(y-y_0)^2\) . If \(c=0\)  then
\(\psi _x(x,y)=Cp_{n}(x,y-y_0)+O(r^{n+1}),\) 
where \(p\)  is a harmonic polynomial of two variables \(x\)  and \(y-y_0\)  of order \(n\) , \(n>2\) , which is even with respect to \(y-y_0\)  and vanishes for \(x=0\) . All such harmonic polynomials change sign inside \(D\)  and since \(\psi _x\ge 0\)  inside \(D\) , we have that \(C=0\) . This implies that \(\psi \)  has zero of infinite order at \((0,y_0)\) . By unique continuation property we conclude that \(\psi \)  is identically zero. This contradiction proves that \(\psi _x(x,y)=cx+O(r^2)\)  with \(c>0\) . Similarly, we can prove the second inequality in (REF ). Thus (ii) is proved.
Let us prove (i) for points on \(S\) . Denote by \(K\)  the set of points \((x,y)\in \overline{S}\)  for which \(\psi _y(x,y)=0\) . Clearly \(K\)  is compact and
\(\psi _x(x,y)^2\ge c_*;=\min _{(x,y)\in \overline{S}}|\nabla \psi |^2.\) 
The right-hand side here is positive by (REF ).
If \(\psi _y(x_*,y_*)\ne 0\)  then there is an interval \((x_*-\varepsilon ,x_*+\varepsilon )\) , \(\varepsilon >0\) , where
the curve can be parameterized by \(y=f(x)\)  near the point \((x_*,y_*)\) . Differentiating equation \(\psi (x,f(x))=0\)  with respect to \(x\)  we get the relation (REF ).
Then if \(u(x,y)=0\)  on this curve at a certain point \((x,y)\) , then \(u_\nu (x,y)=0\)  by (REF ). On the other hand by the Hopf lemma \(u_\nu \)  must be different from zero. This contradiction proves (i) for \(S\) . So the assertions (i) and (ii) are proved.
Let us prove (iii). From (REF ) and (REF ) it follows that
\(\psi _x(x,y)=C_0xy+o(r_0^2)\;\;\psi _x(x,y)=C_1(x-\Lambda /2)y+o(r_1^2),\) 
where \(r_0\)  and \(r_1\)  are the distances to \((0,0)\)  and \((\Lambda /2,0)\) , and \(C_0\ge 0\) , \(C_1 \le 0\)  respectively. Let \(C_0=0\) . The the asymptotics has the form
\(\psi _x(x,y)=Cp_n(x,y)+o(r_0^n),\) 
where \(p_n\)  is a harmonic polynomial of degree \(n>2\)  vanishing for \(x=0\)  and for \(y=0\) . Such polynomials change sign in \(D\)  and therefore \(C=0\) , which leads to \(\psi _x(x,y)=O(r_0^n)\)  for any \(n\) . This implies \(\psi _x=0\) . This contradiction proves that \(C_0>0\) . Similarly one can prove that \(C_1<0\) .
